If line `P Q` , where equation is `y=2x+k` , is a normal to the parabola whose vertex is `(-2,3)` and the axis parallel to the x-axis with latus rectum equal to 2, then the value of `k` is `(58)/8` (b) `(50)/8` (c) `1` (d) `-1`

Options

A. `58//8`
B. `50//8`
C. 1
D. `-1`

Explanation

To solve the problem step-by-step, we will follow the information provided in the question and the video transcript.

### Step 1: Identify the equation of the parabola
Given that the vertex of the parabola is at \((-2, 3)\) and the axis is parallel to the x-axis with a latus rectum of 2, we can use the standard form of the equation for a parabola that opens horizontally:
\[ (y - k)^2 = 4a(x - h) \]
Here, \( (h, k) \) is the vertex of the parabola, and \( a \) is half the length of the latus rectum. Since the latus rectum is 2, we have:
\[ a = \frac{2}{4} = \frac{1}{2} \]
Substituting \( h = -2 \) and \( k = 3 \) into the equation:
\[ (y - 3)^2 = 2(x + 2) \]

### Step 2: Write the equation of the normal to the parabola
The equation of the normal to the parabola at a point can be expressed as:
\[ y - k = m(x - h) \]
Where \( m \) is the slope of the normal. We know that the slope of the normal line is given as \( m = 2 \) (from the equation \( y = 2x + k \)).

### Step 3: Substitute the values into the normal equation
Substituting \( m = 2 \), \( h = -2 \), and \( k = 3 \) into the normal equation:
\[ y - 3 = 2(x + 2) \]
Expanding this:
\[ y - 3 = 2x + 4 \]
\[ y = 2x + 7 \]

### Step 4: Compare with the given normal line equation
The given normal line equation is \( y = 2x + k \). From our derived equation \( y = 2x + 7 \), we can see that:
\[ k = 7 \]

### Step 5: Find the value of \( k \)
Now we need to find the value of \( k \) in the context of the options provided. Since the options are:
- (a) \( \frac{58}{8} \)
- (b) \( \frac{50}{8} \)
- (c) \( 1 \)
- (d) \( -1 \)

We can convert \( k = 7 \) into a fraction:
\[ k = \frac{56}{8} \]
This value does not match any of the options provided.

### Conclusion
It appears there may have been an error in the interpretation of the problem or the options given. However, based on the calculations, the derived value of \( k \) is \( 7 \) or \( \frac{56}{8} \).